The article examines US inflation and economic activity using extended Phillips Curve models. The researchers find that removing low frequency movements in the data can lead to poor predictions. They incorporate forward and backward-looking inflation expectations and use survey data to improve predictions. The study shows no evidence of a long-term relationship between inflation and marginal costs. Backward-looking inflation is more influential than forward-looking inflation. The extended models provide more precise estimates of inflation levels and volatilities compared to other models. The researchers suggest an increased probability of deflation in recent years based on the tails of the predictive distributions.